Popular Science Monthly/Volume 58/March 1901/Chapters on the Stars IX
|CHAPTERS ON THE STARS.|
STATISTICAL STUDIES OF PROPER MOTIONS.
The number of stars now found to have a proper motion is sufficiently great to apply a statistical method to their study. Several important steps in this study have been taken by Kapteyn, who, in several papers published during the past ten years, has shown how conclusions of a striking character may be drawn in this way.
We must begin our subject by showing the geometrical relations of the proper motion of a star, considered as an actuality in space, to the
proper motion as we see it. The motion in question is supposed to take place in a straight line, with uniform velocity. Leaving out the rare cases of variations in the motion due to the attraction of a revolving body, there is nothing either in observation or theory to justify us in assuming any deviation from this law of uniformity. The direction of a motion has no relation to the direction from the earth to the star. That is to say, it may make any angle whatever with that direction.
Let E be the position of our solar system, and S that of a star moving in the direction of a straight line, S D. It must not be stood that the length of this line is taken to represent the actual motion; the latter would be infinitesimal as compared with its length; we use it only to show direction. We may, however, use the line to represent on a magnified scale the actual amount of the motion during any unit of time, say, one year. It may be divided into two components; one, S, in the direction of the line of sight from us to the star, which for brevity we shall call the radial line, and the other, S M, at right angles to that line.
It must be understood that, as the term 'proper motion' is commonly used, only the component S M, can be referred to, because the radial component, S R, does not admit of being determined by telescopic vision. As we know from the preceding chapters, it can in the case of the brighter stars be determined by spectroscopic measurement of the radial motion. At present we leave this component out of consideration.
The visible component, S M, can also be resolved into two perpendicular components, the one east and west on the celestial sphere, the other north and south. The former is the proper motion in right ascension (the measured motion in this coordinate being multiplied by the co-sine of the declination to reduce it to a great circle), and the other is the proper motion in declination. In star catalogues these two motions are given, so far as practicable. Thus, altogether the actual motion of a star in space may be resolved into three components: that of right ascension, that of declension, and the radial component.
An additional consideration is now to be added. The proper motion of a star, as observed and given in catalogues, is a motion relative to our system. It has been shown in a former chapter that the latter has a proper motion of its own. When account is taken of this, and the motions are all reduced as well as we can to a common center of gravity of the whole stellar system, we conceive the observed proper motion of the star to be made up of two parts, of which one is the actual motion of the star relative to the common center, and the other due to the motion of the sun, carrying the earth with it. The direction of the latter appears to us opposite that of the motion of the sun. The sun's motion being directed to the constellation Lyra, it follows that the component in question in the case of the stars is directed toward the opposite constellation, Argo. This component, as we know, is termed the parallactic motion, being dependent on the distance or parallax of the star.
As in the case of other proper motions, we may measure the parallactic motion either in angular measure, as so many seconds per century, or in linear measure, as so many kilometers per second. The relation of the two measures depends on the distance of a star. The simplest conception of the relation may be gained by reflecting that the parallactic motion of a star lying at right angles to the direction of the solar motion during the time that the sun, by its proper motion, is passing over a space equal to the radius of the earth's orbit, is equal to the parallax of the star. For this parallax is simply the angle subtended by that radius as seen from the star; and the same angle is the difference in direction of the star as seen from the two ends of the radius.
As yet, the actual amount of the sun's motion has not been well determined. Kapteyn's estimate is 16.7km. per second, which may be called 10 miles. But the results of additional determinations of radial motions make it likely that this result should be increased to perhaps 19 or 20km. per second, or 4 radii of the earth's orbit per year. Accepting this speed we shall have the following rule:
The parallax of a star lying in a direction nearly at right angles to that of the solar motion is equal to one-fourth of its parallactic motion in a year.
In the case of stars in other directions, the parallax would be greater in proportion to the cosecant of the angle between the direction of the star and the solar apex.
If the stars were at rest this rule would enable us immediately to determine the distance of any star by its proper motion, which would then be simply the parallactic motion itself. Unfortunately, in the case of any one star considered individually, there is no way of deciding how much of its motion is proper to itself and how much is the parallactic motion. But when we consider the great mass of stars, it is possible in a rough way to make a distinction between the two motions in a general average.
The direction or motion of any particular star having no reference to that of the sun is as likely to be in the direction of one of the three components we have described as of any other. Hence, in the average of a great number of stars we may conclude that these components are equal.
One of the simplest applications of this law will enable us to compute the mean parallax of the stars whose radial motions have been determined. As this application is, in the present connection, made only for the purpose of illustration, I shall confine myself to the 47 stars of which the radial motions have been measured by Vogel. The mean annual proper motions of these stars, taken without any regard to their signs, are:
|Including Arcturus.||Omitting Arcturus.|
|In right ascension||0.163||0.144|
The difference of the mean motions in right ascension and declension is to be regarded as accidental. The velocity of Arcturus is so exceptionally great that we ought, perhaps, to leave it out in taking the mean.
Now, the mean of the radial motions as found by Vogel is 16 kilometers per second. By hypothesis the actual motion in the radial line is in the general average the same as in the other two directions. We have, therefore, to acquire what must be the parallax of a star in order that, moving with a velocity of 16 kilometers per second, its angular proper motion may have one of the above values. This result is by a simple computation found to be:
|From the mean motion in R.A.||0.049||or||0.043|
|From the mean motion in Dec.||0.064||or||0.035|
The difference of these results shows the amount of uncertainty of the method. Our general conclusion, therefore, is that the mean parallax of the Vogel stars, which may be regarded as corresponding approximately to the mean parallax of all the stars of the second magnitude, is about 0".04.
We have spoken of the two components of the apparent motion as those in right ascension and declination, respectively. But there is no particular significance in the direction of these coordinates, which have no relation to the heavens at large. For some purposes it will be better to take as the two directions in which the motions are to be resolved that of the parallactic motion and that of right angles to it. That is to say, taking the solar apex as a pole, we conceive a line drawn from it to the star, and resolve the apparent motion upon the celestial sphere into two components, the one in the direction of this line, the other at right angles to it. The former, which we may call the apical motion, is affected by the parallactic motion; the latter, which we call the cross-motion, is not, and therefore shows the true component of the motion of the star itself in the direction indicated.
Kapteyn has gone through the labor of resolving all the proper motions of the Bradley stars given by Auwers, in this way. His assumed position of the solar apex was:
|Right ascension||276° = 18h. 24m.|
The radically new treatment found in this paper embraces three points. The first consists in the distinction between the spectral types of the different stars and the separate study of the proper motions peculiar to each type. The next point is the reference of the motions to the solar apex. The third is the study of the relations of the stars to the galactic plane.
A remarkable relation existing between the spectral type of stars and their proper motions was brought out by these investigations. The stars of Type I. have, in the general mean, smaller proper motions than those of Type II. The following table is made up from Kapteyn's work. First we give the limits of proper motion; then on the same line the number of stars of the respective Types I. and II. having proper motions within these limits:
|Centennial||Number of Stars.|
|Prop.||motions.||Type I.||Type II.|
It will be seen that in the case of stars having proper motions of less than 5" per century a large majority are of Type I. In the case of proper motions between 6" and 9" the number is nearly equal. Between 10" and 20" there is a large majority of Type II. Between 30" and 49" the number of Type II. is nearly five times that of Type I. Finally, only three stars of Type I. have proper motions exceeding 50", while 58 stars of Type II. have a proper motion exceeding this limit.
We may make two hypotheses on this subject: one, that the stars of Type II. really move more rapidly than those of Type I.; the other, that their actual motion is the same, but that the stars of Type I. are more distant stars. The last conclusion seems much more probable, and is strengthened by the much greater condensation of stars of Type I. toward the Milky Way.
Let us now consider the principles by which we may study a great collection of proper motions statistically. There are scattered around us in the stellar spaces, in every direction from us, a large number of stars, each moving onward in a straight line and in a direction which, with rare exceptions, has nothing in common with the motion of any other star. The velocities of the motion vary from one star to another in a way that can not be determined, some moving slowly and some rapidly. Is it possible from such a maze of motions to determine anything? Certainly we can not learn all that we wish, yet we may learn something that will help us form some idea of the respective distances of the stars and the actual velocity of their motions. An obvious remark is that the more distant a star the slower it will seem to move. We must, therefore, distinguish between the linear or actual motion of a star, expressed as so many kilometers per second, and its apparent or angular motion of so many seconds per year, derived by measuring its change of direction as we see it with our instruments.
We shall now endeavor to explain Kapteyn's method in such a way that the reasoning shall be clear without repeating the algebraic operations which it involves. Let us conceive that Fig. 2 is drawn on the celestial sphere as we look up at the heavens. S is the direction of a star in the sky as we see it. Let us also suppose that the solar apex, situated in the constellation Lyra, lies anywhere horizontally to the left of the star, in the direction of the arrow-head marked Apex. Suppose
also that, were the solar system at rest, we should see the star moving along the line S D. Let the length of the line S D represent the motion in some unit of time, say, one year. Next, suppose the star at rest. Then in consequence of the motion of the solar system, by which we are carried toward the apex, the star would seem to be moving with its parallactic motion in the direction S B, away from the apex. Let the length of this line represent the parallactic motion in one year. Then by the theory of composition of motions, the star moving by its real motion from S to D, and by the motion of the earth having an apparent motion from S to B, will appear to us to move along the diagonal S A of the parallelogram. Thus, the line S A will represent the annual proper motion of the star as we observe it with our instruments, and which can be resolved into the apical motion, in the direction S B, and is cross-motion in the direction Sτ.
The apical motion consists of two parts, one the parallactic motion, equal to S B; the other real, and due to the motion of the star itself along the line S D, and equal to the distance of D from the line Sτ.
We have now to inquire how, in the case of a great number of stars, we may distinguish between the two parts.
We now make the general hypothesis that, in the average of a great number of stars, actual motions have no relation to the direction of our sun from the star. Then the components of the actual motion, S D, will in the general average have equal values, positive and negative canceling each other. Hence, if we take the mean of a great number of motions along the apical line it will give us the value of S B due to the motion of the earth, and, hence, the mean parallactic motion of all the stars considered.
The problem now becomes one of averages. We wish to form at least a rude estimate of the average speed of a star in miles or kilometers per second. To show how this may be done let us suppose that we observe the proper motions of a great number of stars at some distance from the solar apex, so that their parallactic motion shall be observable. Stumpe and Ristenpart, the German astronomers, as well as Kapteyn, have considered the relation between the two motions in the following way: We divide the stars observed into classes, taking, say, one class having small, but easily measured, proper motion; another having a proper motion near the average, and a third, of large proper motion. Sometimes a fourth class is added, consisting of stars having exceptionally large proper motions. From each of these classes we can determine, as already shown, the average motion from the direction of the solar apex; that is to say, the average parallactic motion. This will be inversely as the average distance of the stars.
Stumpe's three classes were: I., proper motions ranging from 16" to 32" per century; II., between 32" and 64" per century; III., between 64" and 128" per century; IV., greater than 128". The average of the proper motions in each class, the average of the parallactic motions and the ratio of the two are these:
|Class||Prop. Mot.||Par. Mot.||Quotient|
It will be seen that the ratio of the proper motion of the star to the parallactic motion diminishes as the former increases.
The same thing was found by Ristenpart from the proper motions of the Berlin zone, as shown below:
|Class||Prop. Mot.||Par. Mot.||Quotient|
The smaller value of the quotient from stars near to us than from the more distant stars was supposed to lead to the conclusion that the latter had a more rapid real motion than the former. A little thought will show that, while this is quite true of the stars included in the list, this does not prove it to be true for the stars in general. We can not, as already pointed out, determine the motion of any star unless it exceeds a certain limit. Hence, in the case of the more distant stars we can observe the proper motions only of those which move most rapidly, while in the case of the nearer ones we may have measured them all. We should, therefore, naturally expect that the more distant stars in our list will show too large a value of the proper motion, for the simple reason that those having small proper motion are not included in the average. There is, therefore, no evidence that the more distant stars move faster than the nearer ones.
An error in the opposite direction occurs through the method of selecting stars of given proper motion. We have already pointed out that in the case of any individual star we cannot determine how much of its apparent apical motion may be that of the star itself, and how much the parallactic motion arising from the motion of the earth. What we have done is to assume that in the case of a great number of stars the actual apical motions will be equal, and in the opposite directions, so as to cancel each other in the average of a great number, leaving this average as the parallactic motion. Now, to fix the ideas, suppose that two stars have an equal apical motion, say 3 radii of the earth's orbit in a year, but in opposite directions. The apical motion of the earth being 4 radii per year, it follows that the star which is moving in the same direction as the earth will have a relative apical motion of only 1, and will, therefore, not appear in our list as a star of large proper motion. On the other hand, the star moving with equal speed in the opposite direction will have a motion of 7 radii per year, and, will, therefore, be included among stars of considerable proper motion. Thus, a bias occurs, in consequence of which we include many stars having a motion away from the solar apex, while the corresponding ones, necessary to cancel that motion, will be left out of the count. Thus, the parallactic motion will, in the average, be too large in the case of the stars of large apparent proper motion. Now, this is exactly what we see in the above tables. As we take the classes with larger and larger proper motions, the supposed parallactic motion, which is really the mean of the apical motions, seems to increase in a yet larger degree. It is, therefore, impossible to determine from comparisons like these what the exact ratio is.
This error is avoided when we do not arrange and select the stars according to the magnitude of their proper motions, but take a large list of stars, determine their proper motions as best we can and draw our conclusions from the whole mass. This has been done by Kapteyn in the paper already quoted; and by a process too intricate to be detailed in the present work he has reached certain conclusions as to the ratio of the actual motion of the sun in space to the average motion of the stars. His definitive result is:
Average speed of a star in space
= Speed of solar motion 1.86.
This I shall call the straight-ahead motion of the star, without regard to its direction. But the actual motion as we see it is the straight-ahead motion, projected on the celestial sphere. The two will be equal only in cases where there is no radial motion to or from the earth. In all other cases the motion which we observe will be less than the straight-ahead motion. By the process of averaging, Kapteyn finds:
Linear projected speed of a star
= Speed of solar motion 1.46.
This projected motion, again, may be resolved into two components at right angles to each other. It follows that the average value of either component will be less than that of the projected motion. The components may be the motions in right ascension or declination, or the apical motion and the motion at right angles to it. In any case, the mean value of a component will be:
Speed of solar motion 0.93.
I have used Kapteyn's numbers to obtain the same relation by a somewhat different and purely statistical method.
Imagine the proper motion of a star situated nearly at right angles to the direction of the solar motion. Although we cannot determine how much of its apical motion is actual and how much is parallactic, we can determine whether its motion, if toward the solar apex, exceeds that of the sun. In fact, all stars the apical component of whose motion is in the same direction and greater than that of the sun, whatever the distance of the star, appear to us as moving toward the apex, a direction to which we assign a negative algebraic sign. All stars moving more slowly than this, or in the opposite direction from the sun, will have apparent motions away from the apex, which we regard as algebraic positive. We can, therefore, by a simple count separate the stars moving in the same direction as the sun, and with greater speed, from all the others.
I have classified the stars in this way not only as a whole, but also with reference to their cross-motion—motion at right angles to that of the sun. That is to say, I have taken the stars whose cross-motion, τ, is 2" per century or less and counted their apical motions as positive, negative and zero. Then, I have done the same thing with cross-motions of 3" or 4", then with cross-motions ranging from 5" to 7", and so on. All cross-motions above 13" we put together. The results of this work are shown, so far as described, in the first four columns of the table below. We have here for the various values of τ the number of positive, negative and zero apical motions.
Table showing the number of positive and negative apical motions for different values of the cross-motion.
|Values of τ||Special Motions, σ||Percentage.|
|0, ± 1, 2||1,013||261||425||1,143||555||0.33||0.67|
|± 3, 4||360||56||160||388||188||0.33||0.67|
|± 5 to 7||285||37||107||303||125||0.29||0.71|
|± 8 to 12||215||7||52||218||55||0.20||0.80|
|± 13 +||216||2||61||217||62||0.22||0.78|
The first question that arises in connection with this table is how to count the motions that come out zero; that is to say, those which are too small to be certainly observed. The most probable distribution we can make of them is to suppose that they are equally divided between positive and negative motions. I have, therefore, added one-half of the zero motions to the positive and one-half to the negative column, thus getting the results given in columns P and N. The percentages of positive and negative motions thus resulting are given in the last column.
We see that there is a fairly regular progression in the percentage, depending on the value of the cross-motion. In the case of the small cross-motions, which presumably belong to the more distant stars, the percentage of negative motions is markedly greater than it is in the case of the nearer stars which have larger values of τ. The diminution in the number of zero motions is still more remarkable. This arises from the fact that in the case of the nearer stars the apical motions are necessarily larger, whether positive or negative.
In the preceding table all the stars were counted, without reference to their distance from the solar apex. The result of this will be that the mean of the apical motions is taken as we see it projected on the sphere, which does not correspond to the actual motion in space except when the direction of the star is at right angles to that of the apex. I have, therefore, made a second partial count, including only stars between 60° and 120° from the apex. These stars were selected in opposite regions of the heavens, so as to eliminate any constant error depending on the right ascension. The result of a count of 733 stars is:
If we proceed as before, dividing the zero motions equally between the positive and negative ones, we shall find the respective numbers to be 555 and 178. The percentage of negative motions is, therefore, 24. This will still be slightly too large, owing to the obliquity under which many of the stars were seen. We may estimate the most likely percentage as 23.
We conclude, therefore, that when the motions of all the stars are so resolved that one component shall be that in the direction of the apex, 23 per cent, of the stars will be found moving toward the apex with a greater speed than that of the sun. It may, therefore, be assumed that in the general average an equal number are moving in the opposite direction with a greater speed than that of the sun. We conclude, therefore, that the resolved motion of 46 of the stars is greater than of the sun, leaving 54 per cent. less.
In the absence of an exact knowledge of the relation between the magnitude and the number of motions, we shall not be far wrong in assuming that one-half the stars move to or from the apex with more than the average speed, and one-half with less. Comparing this with the percentage found, we may conclude that the average motion of a star is less than that of the sun, in the ratio 46:50; or that it is found by multiplying the motion of the sun by the factor 0.92. This is almost exactly the number which we have quoted from Kapteyn.
We have already stated that the actual speed of the solar motion, still somewhat uncertain, may be estimated at 20 kilometers per second, or 4 radii of the earth's orbit in a year. For our present purposes the latter method of expressing the velocity is the more convenient. Multiplying this speed by the factors already found, we have the following results for the average proper motions of a star in space expressed in kilometers per second, and radii of the earth's orbit in a year:
|Motion in one component||19km.||=||3.7r.|
The motion of 20km. or 4r. assigned to the sun is its straight-ahead motion. This is little more than half the average. It follows that our sun is a star of quite small proper motion.
We shall now bring the lines of thought which we have set forth in the preceding chapters to converge on our main and concluding problem, that of the distribution of the stars in space. While we cannot reach a conclusion that can claim numerical exactness, we may reach one that will give us a general idea of the subject. The first question at which we aim is that of the number of stars within some limit of distance. It is as if, looking around upon an extensive landscape in an inhabited country, we wished to estimate the average number of houses in a square mile. On the general average, what is the radius of the sphere occupied by a single star? If we divide the number of cubic miles in some immense region of the heavens by the number of stars within that region, what quotient should we get? Of course, cubic miles are not our unit of measure in such a case. It will be more convenient to take as our unit of volume a sphere of such radius that from its center, supposed to be at the sun, the annual parallax of a star on the surface would be 1". The radius of this sphere would be 206,265 times that of the earth's orbit. We may use round numbers, consider it 200,000 of these radii, and designate it by the letter E.
Now, let us conceive drawn around the sun as a center concentric spheres of which the radii are R, 2R, 3R, and so on. At the surfaces of these respective spheres the parallax of a star would be 1", half a second, one-third of a second, and so on. The volumes of spheres being as the cubes of their radii, those of the successive spheres would be proportional to the numbers 1, 8, 27, 64, etc.
If the stars are uniformly scattered through space, the numbers having parallaxes between the corresponding limits will be in the same proportion.
The most obvious method of determining the number of stars within the celestial spaces around us is by measurement of their parallaxes. It is possible to reach a definite conclusion in this way only in the case of parallaxes sufficiently large to be measured with an approach to accuracy. In the case of a small parallax the uncertainty of the latter may be equal to its whole amount. In this case the star may be at any distance outside the sphere given by its measured parallax, or far within that sphere, so that no conclusion can be drawn. It is, on the whole, useless to consider parallaxes less than 0".10; even those having this value are quite uncertain in most of the cases. The data at command for our purpose are the known individual parallaxes and the statistical summary given by Dr. Chase as the result of his survey, and quoted in our chapter on the parallaxes of the stars. This survey was confined to stars whose parallax was not already measured, and it brought out no parallax exceeding 0".30.
The most careful search has failed to reveal any star with a parallax as great as 1", and it is not likely that any such exists. It is, therefore, highly probable that the first sphere will not contain a single star except the sun in its center.
Within the third sphere, the parallax at the surface of which is 0".33, we may place the following for stars with entire certainty:
There are two other cases in which the parallax is doubtful, though the measures as made bring the stars within the sphere 3R. They are:
|O. A. 18,609||0.35|
In the case of η Herculis the proper motion is so small that the presumption is strongly against so large a parallax, and the doubtful parallax of the last star is so near the limit that it may be left out of the count. The doubt in its case may be set off against a doubt whether the parallax assigned to LI. 221,185 is not too large. We assume, therefore, that four stars are contained within the sphere 3R, the volume of which is 33 = 27. This would give, in whole numbers, one star to 7 unit spheres of space.
When we come to smaller parallaxes we find a great deficiency in the number measured in the Southern hemisphere. The policy of Gill, under whose direction or with whose support all the good measures in that hemisphere were made, was to make a few very thorough determinations rather than a general survey. Between the limits 0".20 and 0".33 are found:
|In the||Southern||Hemisphere||4||meas. (Gill)|
Of the Northern results three are exactly on the limit, 0".20, and several others are doubtful, and probably too large. The most likely number for the Northern hemisphere seems to be 12, and if we estimate an equal number for the Southern hemisphere we shall have 24 in all. Adding the four stars within the sphere 3R, we shall then have a total of 28 within the sphere 5R, of which the volume is 125. This gives between 4 and 5 space units to a star.
Let us now consider the space between the spheres 5R and 10R, including all stars whose parallax lies between the limits 0".10 and 0".20. Of these the numbers are:
Reasoning as before, we may assume that the number of stars between the assigned limits is 60, making a total of 88 within the sphere 10R. The volume of space enclosed being 1,000 units, this will give one star to 12 units of space.
How far can we rely on this number as an approximation to the actual number of stars within the tenth sphere? The errors in the estimate are of two classes, those affecting the parallax itself and those arising from a failure to include all the stars within the sphere. The very best determinations are liable to errors of two or three hundredths of a second, the inferior ones to still larger errors. Thus, it may happen that there are stars with a real parallax larger than the limit of which the measures fall below it and are not included, and others smaller than the limit which, through the errors of measurement, are made to come within the sphere. As we have seen in the chapter on the parallaxes, it is quite possible that there may be a number of stars with a measurable parallax whose proximity we have never suspected on account of the smallness of the proper motion. We can only say that the nearer a star is to the system the more likely its proximity is to be detected, so that we are much surer of the completeness of our list of large parallaxes than of small ones. Hence, there may well be a number of undetermined parallaxes upon or just above the limit 0".10.
The most likely conclusion we can draw from this examination seems to be that in the region around us there is one star to every 8 units of space; or that a sphere of radius, 2R, equal to 412,500 radii of the earth's orbit, corresponding to a parallax of 0".50, contains one star. This is a distance over which light would pass in 8½ years.
We next see how far a similar result can be derived from statistics of the proper motions. It seems quite likely that nearly all proper motions exceeding 1" annually have been detected. The number known is between 90 and 100, but it can not be more exactly stated because there is some doubt in the case of a number which seem to be just about on the limit. In this value, 1", is included the effect of the parallactic motion, which, on the general average, increases the apparent proper motion of a star. To study this effect let us call the list of 90 or more stars actually found List A. Were it possible to observe the proper motions of the stars themselves separate from the parallactic motion, we should find that, when we enumerate all having a proper motion of more than 1", we should add some to our List A and take away others. The stars we should add would be those moving in the same direction as the sun, whose motions appear to us to be smaller than they really are, while we should take away those moving in the opposite direction, whose motions appear to us larger than they really are. On the average, we should take away more than we added, thus diminishing slightly the number of stars whose motion exceeds 1". Making every allowance, we may estimate that probably 80 stars have an actual proper motion on the celestial sphere of 1" or more. We have found that the average linear proper motion of a star, as projected on the sphere, is about 6 radii of the earth's orbit annually. A star having this motion would have to be placed at the distance 6R to have, as seen by us, an angular motion of 1". The parallax corresponding to the surface of this sphere is 0".167. The volume of the sphere is 216, and according to our estimate from the parallaxes it would contain only 27 stars. It will be seen that these results give a greater density of the stars than the result from the measured parallaxes; that is to say, they indicate that there are still an important number of measurable parallaxes to be determined, while the number of stars is less than would be derived from their proper motions. But the fact is that the number of stars estimated as within a given sphere by the proper motions will be in excess, owing to the actual diversity of these proper motions, which may range from to a value several times greater than the average. In consequence of this, our list of stars with a proper motion exceeding 1" will contain a number lying outside the sphere 6R, but having a proper motion larger than the average. We are also to consider that within the sphere may actually lie stars having a proper motion less than the average, which will, therefore, be omitted from the list. Of the number of omitted and added stars the latter will be the greater, because the volumes of spheres increase as the cubes of their radii. For example, the space between the spheres 6R and 9R is more than double that within 6R, and our list will include many stars in this space. The discrepancy between the parallaxes and the proper motions probably arises in this way.
Let us see what the result is when we take stars of smaller proper motion. The most definite limit which we can set is 10" per century. We have seen that Dr. Auwers, in his zone, found 23.9 stars per 100 square degrees having a proper motion of 10" or more. This ratio would give about 10,000 for the whole heavens. The sphere corresponding to this limit of proper motion is 60R. On our hypothesis as to star density this sphere would contain 27,000 stars, nearly three times the number derived from Auwers's work. But it is not at all unlikely that even this sphere in question contains twice as many stars as have been detected. Great numbers of the more distant stars will not have been catalogued, owing to their faintness, because a star at the distance 60R will shine to us with only one per cent, the light of one at distance 6R. This corresponds to a diminution of five magnitudes; that is to say, a star of the sixth magnitude at distance 6R would only be of the eleventh magnitude at distance 60R, and would, therefore, not be catalogued at all. There is, therefore, no reason for changing our estimate of star density, which assigns to each star around us 8 units of volume in space.
This fact suggests another important one. Owing to the great diversity in the absolute magnitude of the stars, those we can observe with our telescopes will naturally be more crowded in the neighborhood of our system than they will at greater distances.
Some further results as to the mean parallax of the stars may be derived from a continuation of the statistical study of the proper motions. Kapteyn's investigation in this direction includes a determination of the mean parallactic motion of the stars of each magnitude for the first and second spectral types separately. From this he obtains the following mean parallaxes for stars of the different magnitudes:
Mean parallaxes of stars of different magnitudes, and of the two principal types, as found from their parallactic motions:
|Mag.||Type I.||Type II.|
Unfortunately, owing to the great diversity in the absolute brightness of the stars, and the resulting great difference in the distances of stars having the same magnitude, these numbers can give us only a vague idea of the actual parallaxes. Let us take, for example, the stars of the sixth magnitude. A few of these are, doubtless, quite near to us and have a parallax several times greater than that of the table. Excluding these from the mean, an important fraction of the remainder will have a parallax much smaller than that of the table.
We get a slightly more definite result by studying another feature of the proper motions. We may consider the Bradley stars, whose motions have been investigated, as typical, in the general average, of stars of the sixth magnitude. By a process of reasoning from the statistics, of which I need not go into the details at present, it is shown that the parallactic motion of a large number of these stars, probably one-eighth of the whole, is about 1" per century or less. To this motion corresponds a parallax of 0".0025, corresponding to the sphere of radius 400R.
The statistics of cross-motions lead to a similar conclusion. One-half the Bradley stars have a cross-motion of less than 2 ".5 per century. To this motion would correspond a sphere of radius 200R and a parallax of 0".005. Stars at this distance must be hundreds of times the absolute brightness of the sun to be seen as of the sixth magnitude. Yet the conclusion seems unavoidable that the sphere of lucid stars extends much beyond 400R.
Granting the star density we have supposed, a sphere of radius 400R would contain 8,000,000 stars. As we see many more than this number with the telescope, we have no reason to suppose the boundary of the stellar system, if boundary it has, to be anywhere near this limit.
All the facts we have collected lead to the belief that, out to a certain distance, the stars are scattered without any great and well-marked deviation from uniformity. But the phenomena of the Milky Way show that there is a distance at which this ceases to be true. Let S be the sun, R a portion of the surface of the outer sphere of uniform distribution, and R2 and R3 two contiguous spheres passing through the galactic region G, of which the pole is in the direction P. It is quite certain that the star-density is greater around G than around P. This may arise either from the density at G being greater, or from that at P being less, than the density within the sphere R. From the enormous number of stars collected in the galactic regions, we can scarcely doubt that the former alternative is the correct one. But there must be a sphere at which the second alternative is also correct, because we find the number of stars, even of the lucid ones, to continuously increase from P toward G.
Can we form any idea where this difference begins, or what is the nearest sphere which will contain an important number of galactic stars? A precise idea, no; a vague one, yes. We have seen that the galactic agglomerations contain quite a number of lucid stars, and that, perhaps, an eighth of these stars are outside the sphere 400R. We may, therefore, infer that the Milky Way stars lie not immensely outside this sphere. More than this, it does not seem possible to say at present.
So far as we can judge from the enumeration of the stars in all directions, and from the aspect of the Milky Way, our system is near the center of the stellar universe. That we are in the galactic plane itself seems to be shown in two ways: (1) the equality in the counts of stars on the two sides of this plane all the way to its poles, and (2) the fact that the central line of the galaxy is a great circle, which would not be the case if we viewed it from one side of its central plane.
Our situation in the center of the galactic circle, if circle it be, is less easily established, because of the irregularities of the Milky Way. The openings we have described in its structure, and the smaller density of the stars in the region of the constellation Aquila, may well lead us to suppose that we are perhaps markedly nearer to this region of its center than to the opposite region; but this needs to be established by further evidence. Not until the charts of the international photographic survey of the heavens are carefully studied does it seem possible to reach a more definite conclusion than this.
One reflection may occur to the thinking reader as he sees these reasons for deeming our position in the universe to be a central one. Ptolemy showed by evidence which, from his standpoint, looked as sound as that which we have cited, that the earth was fixed in the center of the universe. May we not be the victims of some fallacy as he was?
- This work of Kapteyn is yet unpublished. The author is indebted to his courtesy for the manuscript copy, with permission to use it. Kapteyn's researches based on this material are contained in a paper on the 'Distribution of the Stars in Space,' communicated to the Amsterdam Academy of Science, January 28, 1893. An abstract in English is found in 'Knowledge' for June 1, 1893.
- The author believes that Monck, of England, independently pointed out this relation, possibly in advance of Kapteyn.
- The author should say that the greater part of the work on these countings was done with great care and accuracy by Mrs. Arthur Brown Davis, to whom he is so much indebted for help of this kind through the present work.